Math 70 TUFTS UNIVERSITY Feb 19, 2020Linear Algebra Department of Mathematics All sections 1,2,3,4,5

Exam 1

Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devicesmust be turned off and put away during the exam. Unless otherwise stated, youmust show all workto receive full credit. You are required to sign your exam. With your signature you are pledging that you haveneither given nor received assistance on the exam. Students found violating this pledge will receive an F in thecourse.

Problem Point Value Points1 122 123 124 135 126 107 108 129 7

100

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Blank page

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1. (12 points) Consider the matrix A =

1 2 0 1

1 1 1 1

2 3 1 2

.(a) Find all solutions to Ax = 0 and write them in parametric vector form.

(b) Let b =

224

. Determine if q =

1

0

0

0

is a solution to Ax = b.

(c) Again, take b =

224

. Given that p =

0

1

1

0

is a particular solution to Ax = b, write out the

general solution to Ax = b in parametric vector form.

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2. (12 points)(a) Complete the definition of a linear transformation T . A transformation T : Rn → Rm is

linear if for all u and v in Rn and all c ∈ R,

(b) Let v1, ..., vp ∈ Rn. Write the definition of Span{v1, ..., vp}.

(c) Let v1, ..., vp ∈ Rn. Define what it means for the set {v1, ...vp} to be linearly independent.

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3. (12 points) Indicate by shading the appropriate box whether each statement is true or false. Forthis problem you do not need to give reasons.

Let A be a 4× 6 matrix and assume that Ax = b is consistent for all b ∈ R4.

(a) The columns of A are linearly independent. T F

(b) There exists a nonzero vector x whose image under the linear transformation T (x) = Ax

is 0. T F

(c) The linear transformation T (x) = Ax is onto. T F

Let B be a 6× 4 matrix and assume that for some b ∈ R6, Bx = b has a unique solution.

(d) There are infinitely many solutions to Bx = 0. T F

(e) Bx = b is consistent for all b ∈ R6. T F

(f) The columns of B span R6. T F

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4. (13 points) Consider a linear system

x1 + x2 + x3 = k

2x1 + hx2 + 2x3 = 3.

(a) Write the augmented matrix of the following linear system and row reduce it to row ech-elon form.

(b) Find all values of h and k (if there are any) so the linear system above has(i) no solutions

(ii) only one solution

(iii) infinitely many solutions.

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5. (12 points) Consider the following vectors and the sets of vectors.

u1 =

000

, u2 =

100

, u3 =

011

, u4 =

101

, u5 =

111

(i) {u1} (ii) {u1, u2} (iii) {u3, u4}

(iv) {u2, u3, u4} (v) {u3, u4, u5} (vi) {u2, u3, u4, u5}

For this problem you do not need to give reasons.

(a) From the above six sets, identify all sets that are linearly independent.

(b) From the above six sets, identify all sets that span a line in R3.

(c) From the above six sets, identify all sets that span a plane in R3.

(d) From the above six sets, identify all sets that span the entire R3.

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6. (10 points) Let T : R2 → R2 be the transformation defined by

T

([x1

x2

])=

[x2

5x1x2

].

Decide whether T is linear justify your answer in the following way.• If you believe T is linear, prove it using the definition of linear transformation.• If you believe T is not linear, provide a specific counterexample (i.e., using specific numbers)to one of the conditions in the definition of linear transformation.

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7. (10 points) Let S = {v1, v2, v3} be a linearly independent set of vectors in Rn. Prove thatS′ = {v1 + v2, v2, v2 + v3} is also linearly independent.

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8. (12 points)

(a) Consider the standard matrix of a linear transformation: A =

[1 0

0 0

].

1) Draw the image of the figure below under the transformation.2) (Circle one) Is this transformation one-to-one? ( Yes, No )3) (Circle one) Is this transformation onto? ( Yes, No )

x

y

x

y

(b) Consider the standard matrix of a linear transformation: B =

[1/3 0

0 2

].

1) Draw the image of the figure below under the transformation.2) (Circle one) Is this transformation one-to-one? ( Yes, No )3) (Circle one) Is this transformation onto? ( Yes, No )

x

y

x

y

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(c) Consider the standard matrix of a linear transformation: C =

[0 −11 0

].

1) Draw the image of the figure below under the transformation.2) (Circle one) Is this transformation one-to-one? ( Yes, No )3) (Circle one) Is this transformation onto? ( Yes, No )

x

y

x

y

9. (7 points) Find the standard matrix of T : R3 → R4 defined by

T

x1x2x3

=

x1 + x2 + x3

2x2 + x3

−x3x1 + x2 + 7x3

.

Math 70 Exam I February 19, 2020

Name:

Please circle your section

Section 1, Eunice Kim, TTh 10:30–11:45

Section 2, Genevieve Walsh, TTh 1:30–2:45

Section 3, Yanghui Liu, MW 9:00–10:15

Section 4, Curtis Heberle, TTh 12:00–1:15

Section 5, Nathan Fisher, MW 9:00–10:15

I pledge that I have neither given nor received assistance on this exam.

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